Is it possible to use multicast on the public internet? If yes: How? Are special IP addresses required and where do you get them from?
You cannot multicast on the public Internet, but you can multicast across the public Internet to another site by using a tunnel that supports multicast.
Multicast routing is very different from unicast routing, and all the routers in the path of the multicast packets need to have multicast routing configured.
As an end-user, you cannot multicast across the Internet, unless using a tunnel.
As a larger organization, like a video provider or an ISP, it is certainly possible to forward multicast packets across their domain boundary (i.e. across an Internet).
Essentially, to forward multicast packets within your own domain (or Autonomous System, AS), you use the PIM protocol and multicast routing. To forward those multicast packets to another AS (like another ISP), you would need a peering agreement with them and use the Multicast Source Discovery Protocol (MSDP), configured on both ends.
While you won't propagate your multicast across the global Internet, crossing network boundaries with multicast packets is not impossible.
PIM and MSDP are advanced, CCIE-level concepts. Here is a good Cisco white paper about it : https://www.cisco.com/c/en/us/td/docs/ios/solutions_docs/ip_multicast/White_papers/mcst_ovr.html#wp1015335